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Scale Factor and Dilations

Grade 10 · Mathematics · Worksheet 3

  1. Sophia dilates point (6, 11) by scale factor 1/2 from center (1, 1). Find the new coordinates. Answer: ______________
  2. Dilate point (8, 11) by scale factor 4 from the origin Answer: ______________
  3. Liam is an urban planner designing a new public plaza. On his initial blueprint, the plaza is represented by a triangle with vertices at A(3, 1), B(7, 9), and C(13, 5). To create a smaller, detailed inset map of the plaza for a walking tour brochure, he applies a dilation centered at the origin with a scale factor of 3/5. What are the coordinates of vertex A' after this dilation? Answer: ______________
  4. Dilate point (5,10) by scale factor 2 from origin Answer: ______________
  5. An architect is designing a new public library. The original blueprint shows a triangular reading nook with vertices at coordinates A(2, 1), B(6, 1), and C(4, 5). The architect decides to enlarge this space by applying a dilation with a scale factor of 2.5, centered at the origin (0, 0). What are the coordinates of vertex C after this dilation? Answer: ______________
  6. Dilate point (7,12) by scale factor 2 from origin Answer: ______________
  7. A pentagon is drawn on a coordinate plane with vertices at A(0, 0), B(10, 0), C(15, 10), D(5, 20), and E(-5, 10). Emma dilates this pentagon from the origin using a scale factor of 0.4. What are the coordinates of the dilated pentagon's vertices? Express your answer as five ordered pairs: A', B', C', D', E'. Answer: ______________
  8. Liam is designing a logo for his robotics team. He creates a small prototype on graph paper with vertices at A(2, 3), B(6, 3), C(6, 7), and D(2, 7). For the final banner, he applies a dilation centered at the origin with a scale factor of 2.5. What are the coordinates of vertex C' after this dilation? Answer: ______________
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Answer Key & Explanations

Scale Factor and Dilations · Grade 10 · Worksheet 3

  1. Sophia dilates point (6, 11) by scale factor 1/2 from center (1, 1). Find the new coordinates. Answer: (3.5, 6) Solution: Translate the point so the center (1, 1) becomes the origin: (6 - 1, 11 - 1) = (5, 10) Apply the scale factor 1/2: (5 × 1/2, 10 × 1/2) = (2.5, 5) Translate back by adding the center coordinates: (2.5 + 1, 5 + 1) = (3.5, 6) The new coordinates are (3.5, 6).
    Full step-by-step solution

    Step 1: Translate the point so the center (1, 1) becomes the origin: (6 - 1, 11 - 1) = (5, 10) Step 2: Apply the scale factor 1/2: (5 × 1/2, 10 × 1/2) = (2.5, 5) Step 3: Translate back by adding the center coordinates: (2.5 + 1, 5 + 1) = (3.5, 6) The new coordinates are (3.5, 6).

  2. Dilate point (8, 11) by scale factor 4 from the origin Answer: (32, 44) Solution: The original point is (8, 11) and the scale factor is 4 Multiply the x-coordinate by the scale factor: 8 × 4 = 32 Multiply the y-coordinate by the scale factor: 11 × 4 = 44 The dilated point is (32, 44) The answer is (32, 44).
    Full step-by-step solution

    Step 1: The original point is (8, 11) and the scale factor is 4 Step 2: Multiply the x-coordinate by the scale factor: 8 × 4 = 32 Step 3: Multiply the y-coordinate by the scale factor: 11 × 4 = 44 Step 4: The dilated point is (32, 44) The answer is (32, 44).

  3. Liam is an urban planner designing a new public plaza. On his initial blueprint, the plaza is represented by a triangle with vertices at A(3, 1), B(7, 9), and C(13, 5). To create a smaller, detailed inset map of the plaza for a walking tour brochure, he applies a dilation centered at the origin with a scale factor of 3/5. What are the coordinates of vertex A' after this dilation? Answer: (9/5, 3/5) or (1.8, 0.6) Solution: Identify the original coordinates of vertex A: (3, 1). The dilation is centered at the origin with scale factor k = 3/5. Multiply the x-coordinate: x' = (3/5) * 3 = 9/5.
    Full step-by-step solution

    Step 1: Identify the original coordinates of vertex A: (3, 1). Step 2: The dilation is centered at the origin with scale factor k = 3/5. Step 3: Apply the dilation formula: (x', y') = (k * x, k * y). Step 4: Multiply the x-coordinate: x' = (3/5) * 3 = 9/5. Step 5: Multiply the y-coordinate: y' = (3/5) * 1 = 3/5. Step 6: The coordinates of vertex A' after dilation are (9/5, 3/5). As decimals: 9/5 = 1.8, 3/5 = 0.6. The answer is (9/5, 3/5).

  4. Dilate point (5,10) by scale factor 2 from origin Answer: (10,20) Solution: The original point is (5,10) and the scale factor is 2 Multiply the x-coordinate by the scale factor: 5 × 2 = 10 Multiply the y-coordinate by the scale factor: 10 × 2 = 20 The dilated point is (10,20) The answer is (10,20).
    Full step-by-step solution

    Step 1: The original point is (5,10) and the scale factor is 2 Step 2: Multiply the x-coordinate by the scale factor: 5 × 2 = 10 Step 3: Multiply the y-coordinate by the scale factor: 10 × 2 = 20 Step 4: The dilated point is (10,20) The answer is (10,20).

  5. An architect is designing a new public library. The original blueprint shows a triangular reading nook with vertices at coordinates A(2, 1), B(6, 1), and C(4, 5). The architect decides to enlarge this space by applying a dilation with a scale factor of 2.5, centered at the origin (0, 0). What are the coordinates of vertex C after this dilation? Answer: (10, 12.5) Solution: 1. Here, scale factor = 2.5. 2.
    Full step-by-step solution

    Step-by-step solution: 1. Understand the dilation: A dilation centered at the origin (0, 0) with scale factor k multiplies each coordinate of a point by k. Here, scale factor = 2.5. 2. Original coordinates of vertex C: C(4, 5) 3. Apply the dilation formula: For a point (x, y) dilated from the origin with scale factor k, the new coordinates are (k*x, k*y). So for C(4, 5) with k = 2.5: New x-coordinate = 2.5 * 4 New y-coordinate = 2.5 * 5 4. Calculate new x-coordinate: 2.5 * 4 = 10 5. Calculate new y-coordinate: 2.5 * 5 = 12.5 6. Final coordinates: C' = (10, 12.5) Thus, after dilation, vertex C is at (10, 12.5).

  6. Dilate point (7,12) by scale factor 2 from origin Answer: (14,24) Solution: Identify the original point coordinates: (7,12) Identify the scale factor: 2 Multiply the x-coordinate by the scale factor: 7 × 2 = 14 Multiply the y-coordinate by the scale factor: 12 × 2 = 24 Write the new coordinates as an ordered pair: (14,24) The answer is (14,24).
    Full step-by-step solution

    Step 1: Identify the original point coordinates: (7,12) Step 2: Identify the scale factor: 2 Step 3: Multiply the x-coordinate by the scale factor: 7 × 2 = 14 Step 4: Multiply the y-coordinate by the scale factor: 12 × 2 = 24 Step 5: Write the new coordinates as an ordered pair: (14,24) The answer is (14,24).

  7. A pentagon is drawn on a coordinate plane with vertices at A(0, 0), B(10, 0), C(15, 10), D(5, 20), and E(-5, 10). Emma dilates this pentagon from the origin using a scale factor of 0.4. What are the coordinates of the dilated pentagon's vertices? Express your answer as five ordered pairs: A', B', C', D', E'. Answer: (0, 0), (4, 0), (6, 4), (2, 8), (-2, 4) Solution: The original vertices are A(0, 0), B(10, 0), C(15, 10), D(5, 20), E(-5, 10). Step 2: Dilation from the origin with scale factor k = 0.4 means each coordinate is multiplied by 0.4.
    Full step-by-step solution

    Step 1: The original vertices are A(0, 0), B(10, 0), C(15, 10), D(5, 20), E(-5, 10). Step 2: Dilation from the origin with scale factor k = 0.4 means each coordinate is multiplied by 0.4. Step 3: Calculate A': (0 * 0.4, 0 * 0.4) = (0, 0). Step 4: Calculate B': (10 * 0.4, 0 * 0.4) = (4, 0). Step 5: Calculate C': (15 * 0.4, 10 * 0.4) = (6, 4). Step 6: Calculate D': (5 * 0.4, 20 * 0.4) = (2, 8). Step 7: Calculate E': (-5 * 0.4, 10 * 0.4) = (-2, 4). The dilated pentagon's vertices are A'(0, 0), B'(4, 0), C'(6, 4), D'(2, 8), E'(-2, 4).

  8. Liam is designing a logo for his robotics team. He creates a small prototype on graph paper with vertices at A(2, 3), B(6, 3), C(6, 7), and D(2, 7). For the final banner, he applies a dilation centered at the origin with a scale factor of 2.5. What are the coordinates of vertex C' after this dilation? Answer: (15, 17.5) Solution: A dilation centered at the origin multiplies each coordinate of a point by the scale factor. Scale factor given: 2.5 Identify the coordinates of vertex C before dilation.
    Full step-by-step solution

    Step 1: Understand the dilation transformation. A dilation centered at the origin multiplies each coordinate of a point by the scale factor. Scale factor given: 2.5 Step 2: Identify the coordinates of vertex C before dilation. From the problem: C(6, 7) Step 3: Apply the dilation to the x-coordinate. x' = scale factor × original x x' = 2.5 × 6 x' = 15 Step 4: Apply the dilation to the y-coordinate. y' = scale factor × original y y' = 2.5 × 7 y' = 17.5 Step 5: Write the new coordinates of C'. C' = (15, 17.5) Final Answer: (15, 17.5)